Often, yes, if you do it wrong. It’s usually not a problem if the bones of the building get a bit wet before the roof goes on; things dry out easily before you install insulation and wall paneling. Larger buildings are all metal, glass and concrete until the exterior is done, so even less problems with water.

It can, yes. Like what @Smashley said.

It could do.

Yes. Materials made of wood will rot. But if the building that is under construction already has a roof sheathings which are water proof then it may not cause much damage. For big buildings, they are usually made of metals and fibre glasses, it will not cause much damage.